Question

For the purpose of calculating 'Palabhā' (the equinoctial noon shadow), what is the standard height required for the 'Śaṅku' (Gnomon) expressed in 'Aṅgula' (digits)?

• A. Six (Ṣaḍ)
• B. Twelve (Dvādaśa)
• C. Eight (Aṣṭa)
• D. Seven (Sapta)

Hint:

Always remember the phrase "Dvādaśāṅgula Śaṅku". It is a fundamental constant in Indian Astronomy, much like 'g' (9.8) is in physics. Whether you are finding the time or the latitude, the gnomon is always 12 units.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
In Siddhanta Jyotish (Indian Mathematical Astronomy), the 'Śaṅku' (Gnomon) is the most fundamental instrument used for measuring time and determining coordinates. It is a straight, vertical rod placed on a horizontal surface. 'Palabhā' is defined as the length of the shadow cast by this Śaṅku at local noon on the day of the equinox (Viṣuvat). This Palabhā is unique to every latitude and is used to calculate the 'Akṣāṃśa' (latitude) of a place.
Key Formula or Approach:
The ancient Indian astronomers adopted a universal standard for the height of the gnomon to ensure consistency in trigonometric calculations across the subcontinent. The standard height is always fixed at 12 Aṅgulas.

Step 2: Detailed Explanation:
The choice of 12 digits (Aṅgula) for the Śaṅku is not arbitrary. In the sexagesimal system (base 60) used by ancient astronomers like Āryabhaṭa and Varāhamihira, 12 is a highly convenient number. Since a circle has 360 degrees and a day has 60 Ghaṭis, the number 12 fits perfectly into these ratios (12 \(\times\) 5 = 60).

When the Śaṅku is 12 units high, the Palabhā \( (s) \) relates to the latitude \( (\phi) \) as follows:
\[ \tan(\phi) = \frac{s}{12} \]
Or in terms of the Sine function (Jyā) used in Sanskrit texts:
\[ \text{Akṣajyā} = \frac{R \cdot s}{\sqrt{s^2 + 12^2}} \]
where \( R \) is the radius of the Sine-table.

Calculating the Palabhā was the primary way to define the 'spatial location' of a person in the ancient world. For instance, the Palabhā at Ujjain (the ancient prime meridian) was taken as approximately 5 digits and 19 minutes. If the Śaṅku height changed, every astronomical table in the Pañcāṅga would need re-calibration. Therefore, all classical texts like Sūrya Siddhānta, Siddhānta Śirōmaṇi, and Grahalāghavam strictly mandate the 'Dvādaśāṅgula Śaṅku' (12-digit gnomon). This standardization allowed travelers and astronomers from different regions to communicate their locations accurately based on shadow lengths.

Step 3: Final Answer:
The height of the Śaṅku for determining Palabhā is 12 (Dvādaśa) Aṅgulas.